1. Chapter 4 Part 1
Translating to and from Z scores, the standard error of the mean, and confidence intervals around muT
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2. Where we have been: Z scores
If you know the proportion from the mean to the score, then you can easily calculate:
The proportion above or below the score.
The percentile rank equivalent.
The proportion of scores between two Z scores.
The expected frequency of scores between two Z scores

3. Concepts behind Z scores
Z scores represent standard deviations above and below the mean.
Positive Z scores are scores higher than the mean. Negative Z scores are scores lower than the mean.
If you know the mean and standard deviation of a population, then you can always convert a raw score to a Z score.
If you know a Z score, the Z table will show you the proportion of the population between the mean and that Z score.

4. Synonyms
Scores above the mean can be called positive Z scores or high scores. The terms are synonymous.
Similarly, scores below the mean can be called negative Z scores or low scores. The terms are synonymous.

5. Raw scores to Z scores
If we know the mean and standard deviation of a population (mu and sigma), any score in that population can be translated into a Z score:
Z =
score - mean
standard deviation =
X -  

6. Z scores to other scores
Conversely, as long as you know the mean and standard deviation of a population, any Z score can be translated into equivalent score in that population.
Score =  + ( Z *  )

7. Calculating z scores Z = score - mean standard deviation
What is the Z score for someone 6’ tall, if the mean is
5’8” and the standard deviation is 3 inches?
Z =
6’ - 5’8” 3” = 3 = 4 3 =
1.33

8. Production F r e q u n c y units
Z score = ( ) / 50
2100
= -80 / 50 =
Standard
deviations
units
2130
2180
2230
What is the Z score for a daily production of 2100, given
a mean of 2180 units and a standard deviation of 50 units?

9. If you know a Z score, you can determine theoretical relative frequencies and expected frequencies using the Z table.
You often start with raw or scale scores and have to convert them to Z scores.
Scale scores are public relations versions of Z scores. They are Z scores with different means and standard deviations.

10. Concepts behind Scale Scores
Scale scores are Z scores expressed in a different way.
In a sense, the most basic scale score is the Z score itself, with mu = 0.00 and sigma = 1.00.
Raw scores can be converted to Z scores, which in turn can be converted to other scale scores.
And other scale scores can be converted to Z scores, that in turn can be converted to raw scores.

11. You need to memorize these scale scores
Z scores have been standardized so that they always have
a mean of 0.00 and a standard deviation of 1.00.
Other scales that are Z scores in disguise use other means and standard deviations for historic or public relations reasons. Three frequently used scales are:
IQ  =100;  = 15
SAT/GRE  =500;  = 100
Normal scores -  =50;  = 10

12. For example: To solve the problem below, convert an SAT Score of 470 to a Z score, then use the Z table as usual.
Proportion mu to Z for Z score of -.30 = .1179 F r e q u n c y
Z score = ( ) / 100
= -30 / 100
470
Proportion at/below score =
= = 38.21% =
Standard
deviations
score
400
500
600
What percentage of test takers obtain a verbal score of
470 or less, given a mean of 500 and a standard deviation of 100?

13. SAT to percentile – first transform to a Z scores
If a person scores 592 on the SATs, what percentile is she at?
SAT  (X-)  (X-)/ 
Proportion mu to Z = .3212
Percentile = ( ) * 100 = = 82nd

14. Convert to IQ scores to Z scores to find the proportion of scores between two IQ scores.
IQ scores have mu = 100 and sigma = 15. What proportion of the scores falls between 85 and 115?
Z score = ( ) / = -15 / 15 = -1.00
Z score = ( ) / 15 = 15 / = 1.00
Proportion = = .6826
What proportion of the scores falls between 95 and 110?
Z score = ( ) / = -5 / 15 = -0.33
Z score = ( ) / 15 = 10 / = 0.67
Proportion = = .3779

15. NOTICE: Equal sized intervals, close to and further from the mean: More scores close to the mean!
Given mu = 100 and sigma = 15, what proportion of the population falls between 95 and 105?
Z score = ( ) / = -5 / 15 = -.33
Z score = ( ) / 15 = 5 / = .33
Proportion = = .2586
What proportion of the population falls between 105 and 115?
Z score = ( ) / = 5 / 15 = 0.33
Z score = ( ) / 15 = 105/ = 1.00
Proportion = = .2120

16. Percentile equivalents of IQ scores: first translate to Z scores
Convert IQ scores of 120 & 80 to percentiles.
X  (X-)  (X-)/ 
mu-Z = .4082, = = 91st percentile, Similarly 80 = = 9th percentile
Convert an IQ score of 100 to a percentile.
An IQ of 100 is right at the mean and that’s the 50th percentile.

17. Going the other way – Z scores to scale scores
Remember:
Score =  + ( Z *  )

18. Convert Z scores to IQ scores: Individual scale scores get rounded to nearest integer.
Z  (Z*)  IQ= + (Z * )
+2.67

19. You try one. Let’s use normal scores (mu=50, sigma = 10).
Translate a Z score of –2.67 to a normal score.

20. Convert Z scores to normal scores: Individual scale scores get rounded to nearest integer.
Z  (Z*)  Score= + (Z * ) –

21. Tougher problems – like online quiz or midterm

22. If someone scores at the 58th percentile on the verbal part of the SAT, what is your best estimate of her SAT score?

23. Percentile to scale score
If someone scores at the 58th percentile on the SAT-verbal,
what SAT-verbal score did he receive?
58th Percentile is above the mean. This will be a positive Z score. The mean is the 50th percentile. So the 58th percentile is 8% or a proportion of above mu. So we have to find the Z score that gives us a proportion of of the scores between mu and Z.
Look at Column 2 of the Z table on page 54. Closest Z score for area of is 0.20
Z  (Z*)  SAT= + (Z * )

24. Note that the only hard part is finding the Z score.
First, ask how far and which direction you are from mean of the scale score.
Second, use that percentage, expressed as a proportion to enter the Z table, looking in Column 2, 4, or 6 for a similar proportion.
Third, choose the Z score closest to that value
Finally, convert that to a scale score, (remembering on which side of the mean you are).

25. Slightly tougher –below the mean

26. Percentile to scale score
If someone scores at the 38th percentile on the SAT-verbal,
what SAT-verbal score did he receive?
38th percentile is below the mean. This will be a negative Z score. The mean is the 50th percentile. So the 38th percentile is 12% or a proportion of below mu. So we have to find the Z score that gives us a proportion of of the scores between mu and Z.
Look at Column 2 of the Z table on page 54. Closest Z score for area of is Z is negative
Z  (Z*)  SAT= + (Z * )

27. Double translations
On the verbal portion of the Wechsler IQ test, John scores
35 correct responses. The mean on this part of the IQ test is
25.00 and the standard deviation is What is John’s
verbal IQ score?
Raw  (X- ) Scale Scale Scale
score (raw) (raw)  Z   score
Z score = / 6.00 = 1.67
IQ score = (1.67 * 15) = 125